Any advice on building an arch over the entry trench? Doesn't look like it should be too hard to get the left side of the door going, but on the right side, the open side of the form is on the wrong side; any suggestions on how to deal with this?

A lot of people use a large plastic trash can to pack the snow over and then pull it out but this is not an option in the backcountry.
A short strait wall can be built with the open end of the form against the igloo. Doing it this way will leave one side of the tunnel with concave blocks and the other wall with convex blocks. Leaning the walls in a bit helps have somewhat of a correct curve but if the tunnel is short, it will get it's support from being attached to the igloo.
To finish the roof, a wall can be built away from the igloo and then cut up into blocks to put in place for the roof. This will take time if you are using sugar snow as it will need to set some time before hard enough to cut and handle.
Another way that works well enough is for one person to stand stooped over in the door while supporting a foam sleeping pad with his back. Another person packs snow onto the pad. This is hard to do in that the person stooped over must remain very still until the packing is complete. Once the packing is complete, the stooped over person releases the pressure. He never pushes up to pack it "just a bit better" or he will break the entire roof.
It is also possible to hold the inner panel in place while another packs snow onto the wall and against to the wall of the igloo. It's tricky and time consuming.
Myself, I usually build in deep snow and the door is on the downhill side. I dig a hallway to the igloo without a roof. I like not having a tunnel so I can just duck under the 8 inch thick wall and I'm inside the igloo quickly.
I would consider a tunnel though if I were to spend and extended time in a very windy area.

Must remember those tips Ed for the ski club trip. It was quite windy on our last trip and the area we chose was not deep enough to have a deep trench entrance. The wind created a bit of a draft around the side of the Igloo door fabric but on the plus side did give ventilation! I'll post the trip report in the Organisations Section of Forum.
Andy

Andy wrote:Must remember those tips Ed for the ski club trip. It was quite windy on our last trip and the area we chose was not deep enough to have a deep trench entrance. The wind created a bit of a draft around the side of the Igloo door fabric but on the plus side did give ventilation! I'll post the trip report in the Organisations Section of Forum.
Andy

I looked at the pictures from your trip report and the top corners of the door need to be sharp/square instead of radiused so the edge of the door fabric will remain in contact with the snow of the door wall.
Your door is also pretty short, you had no choice, and that means you need to hang the door rather high on the igloo so the bottom of the door isn't laying on the ground. When it is high like that, the door can get blown off to the side some and it won't seal properly. A garbage bag will seal better for short doors like that but they are not nearly as easy to go in and out of like our door with the slit through the center of the door.

How about building up a pile of snow like you would for a quinzee. If you do this once the igloo is tall enough the snow can sinter together by the time you finish the igloo. Depending on the snow you might even be able to stand on it while you finish the top. Then all you need to do is dig out the center of the pile and let your partner out.

Hiatus wrote:How about building up a pile of snow like you would for a quinzee. If you do this once the igloo is tall enough the snow can sinter together by the time you finish the igloo. Depending on the snow you might even be able to stand on it while you finish the top. Then all you need to do is dig out the center of the pile and let your partner out.

I think that would work fine when the tunnel is needed. It would surely help being able to stand on it to finish the top. We used to pile snow on the ledge, when we dug out the door, so we could stand on it.
I use a couple different techniques to pack the top now. The most common and easiest is to use a ski pole's basket to pack across the top of the igloo. The other, which I use when I need to get up to take a look at what I'm doing, is to have a teammate kneel down on one knee and put the other leg/knee against the igloo and the other person can step up on his leg/knee. It helps it the one stepping up is light! Or, use a piece of foam pad to cushion the boot tread on the leg. When I'm building a solo igloo, it is short enough that I can lay on the igloo to pack the top. Sometimes I use a ski pole too but it is easy to see what you are doing with the igloo being so short. I build a seven foot for solo trips or sometimes I'll extend the seven foot pole settup so it makes the 7'11" igloo that is shorter than the 8 footer.
But, back to the tunnel.
I tend to have deeper snow than some and have my door below the floor. I've had the door facing into the wind, when there were high winds, with a tarp for a door and it isn't much of a factor for heat loss. I don't feel the tunnel is really needed.
For those with shallow snow, where the door needs to go up into the igloo wall, a door tunnel would help and this would work well. One would end up digging a tunnel which would make the entryway a little more work but it would certainly be simple to do.
I usually build on a hill side to help the door be below the floor level but that would mean the snow, I use to make the tunnel pile, would need to be shoveled uphill.

My wife and I have done what Hiatus suggested and it works just fine, except we do it after completing the igloo. We just dump several shovels of snow in the entry and above it, pack it down, then wait 30 minutes or so for it to sinter and then carve a tunnel into it. The walls of the archway are pre-made since they are largely made up of the snow ledge that one builds up as the igloo is constructed.

If we don't feel like building an arch, we will sometimes build a windbreak by making an L-shaped wall or mound of snow that juts out from the igloo right next to the entryway.

I've always thought myself to be blessed with the deep snow of the Rockies as to why I don't worry about a door tunnel but in reflecting a bit, not so. I've built in shallow snow conditions too but I am usually able to find some slanted ground and make a flat platform by bringing the snow down from the high side to the low side of the platform. This usually gives me tall enough of a base that my door only goes a few inches above the platform when the door is tall enough to crawl through.
Then one time I was up in the Budtrain region and I went around the platform scooping snow onto the platform. I stood with my back to the platform and used the shovel much like paddling a canoe:
The platform ended up being tall enough so the door was below the sleeping level.